R-help to extra exercise on time series

# Read the data and inform R that (the log-transformed) pasenger numbers are time series data with monthly observations:

airline<-matrix(scan("http://www.math.uio.no/avdc/kurs/STK4900/data/airlines"), byrow=T)

airlinets<-ts(log(airline), start=1947, frequency=12)

# Before we look at the questions in the exercise, we will make some plots to get an overview of the data:

# Plot of the data (cf page 106 in BS)

ts.plot(airlinets,ylab="log(passenger)")

# Decompose the time series in trend, sesonal variation and residuals and plot the components:

# (It is beyond the scope of our presentation to describe how this is done. See help(stl) for details.)

airlinets.stl<-stl(airlinets[,1],"periodic")

plot(airlinets.stl)

# Then we turn to the questions in the exercise

# QUESTOINS a&b)

# Define dummy variables for each of the months:

m1<-c(1,rep(0,11))

m2<-c(0,1,rep(0,10))

m3<-c(rep(0,2),1,rep(0,9))

m4<-c(rep(0,3),1,rep(0,8))

m5<-c(rep(0,4),1,rep(0,7))

m6<-c(rep(0,5),1,rep(0,6))

m7<-c(rep(0,6),1,rep(0,5))

m8<-c(rep(0,7),1,rep(0,4))

m9<-c(rep(0,8),1,rep(0,3))

m10<-c(rep(0,9),1,rep(0,2))

m11<-c(rep(0,10),1,0)

m12<-c(rep(0,11),1)

m1<-rep(m1,12)

m2<-rep(m2,12)

m3<-rep(m3,12)

m4<-rep(m4,12)

m5<-rep(m5,12)

m6<-rep(m6,12)

m7<-rep(m7,12)

m8<-rep(m8,12)

m9<-rep(m9,12)

m10<-rep(m10,12)

m11<-rep(m11,12)

m12<-rep(m12,12)

# Check that these commands generate dummy variables as described in the exercise.

# Define a variable for time (i.e. a number for the months)

time<-seq(1,144)

# Fit a linear model without intercept and inspect the results:

mod.a<-lm(log(airline)~m1+m2+m3+m4+m5+m6+m7+m8+m9+m10+m11+m12+time-1)

summary(mod.a)

# Interpret the fitted model and explain why we do not include an intercept term.

# Comment:

# A simpler way to fit the model is by the commands:

month<-rep(1:12,12)

mod.a2<-lm(log(airline)~factor(month)+time-1)

summary(mod.a2)

# Check that these commands give the same results as above and explain why they do so.

# QUESTION c)

# Define centred dummy variables:

mm1<-m1-1/12

mm2<-m2-1/12

mm3<-m3-1/12

mm4<-m4-1/12

mm5<-m5-1/12

mm6<-m6-1/12

mm7<-m7-1/12

mm8<-m8-1/12

mm9<-m9-1/12

mm10<-m10-1/12

mm11<-m11-1/12

# Check that these commands define centred dummy variables as described in the exercise

# Fit a linear model and inspect the results:

mod.c<-lm(log(airline)~mm1+mm2+mm3+mm4+mm5+mm6+mm7+mm8+mm9+mm10+mm11+time)

summary(mod.c)

# Interpret the fitted model.

# Remember to answer the questions in the exercise!

# QUESTION d)

# Plot (the log-transformed) observations and the corresponding fitted values 

plot(time, log(airline))

lines(time, fitted(mod.c)) 

# What does the plot tell you?

# QUESTION e)

# Plot the autocorrelation of the residuals:

acf(ts(resid(mod.c)))

# What does the plot tell you?